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Can someone help explain the passage here? I'm rusty with my linear algebra so the derivate of these transpose matrices isn't making any sense to me. A detailed explanation would be very much appreciated. What happens to the 1/2, and that whole second term in general?

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All you need for this particular question is the following. Let $\mathbf{X}$ be a $T \times K$ matrix, $\mathbf{w}$ a K-dimensional vector and $\mathbf{y}$ a T-dimensional vector, then

$$ \begin{eqnarray*} \frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{y}}{\partial \mathbf{w}}&=& \mathbf{X}^{\prime}\mathbf{y}\\ \frac{\partial \mathbf{w}^{\prime}\mathbf{X}^{\prime}\mathbf{X}\mathbf{w}}{\partial\mathbf{w}}&=&2\mathbf{X}^{\prime}\mathbf{X}\mathbf{w} \end{eqnarray*} $$ Back to your original post, we will first rewrite it in order to apply the rules provided. The first element in the expression we want to differentiate is the same as $\phi'(\mu-R_f\mathbf{1})$, hence $$ \frac{\partial \phi'(\mu-R_f\mathbf{1})-1/2\alpha\phi'\Sigma\phi}{\partial \phi}= (\mu-R_f\mathbf{1})-\alpha\Sigma\phi $$ and equating to 0 you get the desired result.

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